Chapter 2

Chapter 2 - 2 First-Order Differential Equations Exercises...

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y t y x y y x x y y y x x y y x y x x y 2 First-Order Differential Equations Exercises 2.1 1. 2. 3. 4. 5. 6. 7. 8. 17
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y x x y y x x y y x x y -1 0 1 1 2 x 5 4 3 2 1 y 1 2 -1 -2 x 1 y 1 2 -1 -2 x -1 y 1 2 x -5 -4 -3 -2 -1 y Exercises 2.1 9. 10. 11. 12. 13. 14. 15. Writing the diferential equation in the Form dy/dx = y (1 y )(1 + y ) we see that critical points are located at y = 1, y = 0, and y = 1. The phase portrait is shown below. (a) (b) (c) (d) 18
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-1 0 1 1 2 x 5 4 3 2 1 y 1 2 -1 -2 x 1 y 1 2 -1 -2 x -1 y -1 -2 x -5 -4 -3 -2 -1 y 03 1 0 2 -2 5 -2 0 2 024 Exercises 2.1 16. Writing the diferential equation in the Form dy/dx = y 2 (1 y )(1 + y ) we see that critical points are located at y = 1, y = 0, and y = 1. The phase portrait is shown below. (a) (b) (c) (d) 17. Solving y 2 3 y = y ( y 3) = 0 we obtain the critical points 0 and 3. ±rom the phase portrait we see that 0 is asymptotically stable and 3 is unstable. 18. Solving y 2 y 3 = y 2 (1 y ) = 0 we obtain the critical points 0 and 1. ±rom the phase portrait we see that 1 is asymptotically stable and 0 is semi-stable. 19. Solving ( y 2) 2 = 0 we obtain the critical point 2. ±rom the phase portrait we see that 2 is semi-stable. 20. Solving 10 + 3 y y 2 =(5 y )(2 + y ) = 0 we obtain the critical points 2 and 5. ±rom the phase portrait we see that 5 is asymptotically stable and 2 is unstable. 21. Solving y 2 (4 y 2 )= y 2 (2 y )(2 + y ) = 0 we obtain the critical points 2, 0, and 2. ±rom the phase portrait we see that 2 is asymptotically stable, 0 is semi-stable, and 2 is unstable. 22. Solving y (2 y )(4 y ) = 0 we obtain the critical points 0, 2, and 4. ±rom the phase portrait we see that 2 is asymptotically stable and 0 and 4 are unstable. 19
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-2 -1 0 0 ln 9 mg/k mg/k α β α Exercises 2.1 23. Solving y ln( y + 2) = 0 we obtain the critical points 1 and 0. From the phase portrait we see that 1 is asymptotically stable and 0 is unstable. 24. Solving ye y 9 y = y ( e y 9) = 0 we obtain the critical points 0 and ln9. From the phase portrait we see that 0 is asymptotically stable and ln9 is unstable. 25. (a) Writing the diferential equation in the ±orm dv dt = k m ³ mg k v ´ we see that a critical point is mg/k . From the phase portrait we see that mg/k is an asymptotically stable critical point. Thus, lim t →∞ v = mg/k . (b) Writing the diferential equation in the ±orm dv dt = k m ³ mg k v 2 ´ = k m µ r mg k v ¶µ r mg k + v we see that a critical point is p mg/k . From the phase portrait we see that p mg/k is an asymptotically stable critical point. Thus, lim t →∞ v = p mg/k . 26. (a) From the phase portrait we see that critical points are α and β . Let X (0) = X 0 . X 0 , we see that X α as t →∞ .I ± α<X 0 , we see that X α as t . X 0 , we see that X ( t ) increases in an unbounded manner, but more speci²c behavior o± X ( t )a s t is not known.
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Chapter 2 - 2 First-Order Differential Equations Exercises...

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